ON EXPLICIT INTERVAL METHODS OF ADAMS-BASHFORTH TYPE
Jankowska Małgorzata 1, Marciniak A. 1,2
1 Poznań University of Technology, Institute of Computing Science
Piotrowo 3a, 60-965 Poznań, Poland
2Adam Mickiewicz University, Faculty of Mathematics and Computer Science
Umultowska 87, 61-614 Poznań, Poland
e-mail: anmar@sol.put.poznan.pl
DOI: 10.12921/cmst.2002.08.02.46-57
OAI: oai:lib.psnc.pl:535
Abstract:
In our previous paper [1] we have considered implicit interval multistep methods of Adams-Moulton type for solving the initial value problem. On the basis of these methods and the explicit ones introduced by Sokin [2] we wanted to construct predictor-corrector (explicit-implicit) interval methods. However, it turned out that the formulas given by Šokin are incorrect even in the simplest case. Therefore, in this paper we direct our attention to the explicit interval methods of Adams-Bashforth type and modify the formulas of Šokin. For the modified explicit interval methods it is proved, like f o r the implicit interval methods considered in [1], that the exact solution of the problem belongs to interval-solutions obtained by these methods. Moreover, it is shown an estimation of the widths of such interval-solutions.
References:
[1] Jankowska, M., Marciniak, A., Implicit Interval Multistep Methods for Solving the Initial Value Problem, Computational Methods in Science and Technology 8/1 (2002), 17-30.
[2] Šokin, Ju. I.; Interval Analysis [in Russian], Nauka, Novosibirsk 1981.
[3] Conradt, J., Ein Intervallverfahren zur Einschliessung des Fehlers einer Näherungslösung bei Anfangswertaufgaben für Systeme von gewöhnlichen Differentialgleichungen, Freiburger IntervallBerichte 80/1, Universität Freiburg i. Br. 1980.
[4] Eigenraam, P., The Solution of Initial Value Problems Using Interval Arithmetic, Mathematical Centre Tracts Vol. 144, Mathematisch Centrum, Amsterdam 1981.
[5] Hunger, S., Intervallanalytische Defektabschätzung bei Anfangswertauf gaben für Systeme von gewöhnlichen Differentialgleichungen, Schriften der Gesellschaft für Mathematik und Datenverarbeitung Nr. 41, Bonn 1971.
[6] Kalmykov, S.A., Šokin, Ju. I., Juldašev.Z. H., Solving Ordinaiy Differential Equations by Interval Methods [in Russian], Doklady AN SSSR, Vol. 230, No. 6 (1976).
[7] Kalmykov, S. A., Šokin, Ju. I., Juldašev, Z. H., Methods of Interval Analysis [in Russian], Nauka, Novosibirsk 1986.
[8] Marciniak, A., Finding the Integration Intervalfor Interval Methods ofRunge-Kutta Type in Floating-Point Interval Arithmetic, Pro Dialog 10 (2000), 35^15.
[9] Marco witz, U., Fehlerabschätzung bei Anfangswertaufgaben für Systeme von gewöhnlichen Differentialgleichungen mit Anwendung auf das REENTRY-Problem, Numer. Math. 24 (1973).
[10] Marciniak, A.Jnteival Methods ofRunge-KuttaType inFloating-Point Interval Arif/jmerfc [in Polish], Technical Report RB-027/99, Poznan University of Technology, Institute of Computing Science, Poznan 1999.
[11] Moore, R. E., Inteival Analysis, Prentice-Hall, Englewood Cliffs 1966.
[12] Krupowicz, A., Numerical Methods of Initial Value Problems of Ordinaiy Differential Equations [in Polish], PWN, Warsaw 1986.
[13] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, J. Wiley & Sons, Chichester 1987.
[14] Dormand, J. R., Numerical Methods for Differential Equations. A Computational Approach, CRC Press, Boca Raton 1996.
[15] Hairer, E., Nørsett, S. P., Wanner, G., Solving Ordinaiy Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin, Heidelberg 1987.
[16] Matwiejew, N. M., Methods for Integrating Ordinaiy Differential Equations [in Polish], PWN, Warsaw 1982.
In our previous paper [1] we have considered implicit interval multistep methods of Adams-Moulton type for solving the initial value problem. On the basis of these methods and the explicit ones introduced by Sokin [2] we wanted to construct predictor-corrector (explicit-implicit) interval methods. However, it turned out that the formulas given by Šokin are incorrect even in the simplest case. Therefore, in this paper we direct our attention to the explicit interval methods of Adams-Bashforth type and modify the formulas of Šokin. For the modified explicit interval methods it is proved, like f o r the implicit interval methods considered in [1], that the exact solution of the problem belongs to interval-solutions obtained by these methods. Moreover, it is shown an estimation of the widths of such interval-solutions.
[1] Jankowska, M., Marciniak, A., Implicit Interval Multistep Methods for Solving the Initial Value Problem, Computational Methods in Science and Technology 8/1 (2002), 17-30.
[2] Šokin, Ju. I.; Interval Analysis [in Russian], Nauka, Novosibirsk 1981.
[3] Conradt, J., Ein Intervallverfahren zur Einschliessung des Fehlers einer Näherungslösung bei Anfangswertaufgaben für Systeme von gewöhnlichen Differentialgleichungen, Freiburger IntervallBerichte 80/1, Universität Freiburg i. Br. 1980.
[4] Eigenraam, P., The Solution of Initial Value Problems Using Interval Arithmetic, Mathematical Centre Tracts Vol. 144, Mathematisch Centrum, Amsterdam 1981.
[5] Hunger, S., Intervallanalytische Defektabschätzung bei Anfangswertauf gaben für Systeme von gewöhnlichen Differentialgleichungen, Schriften der Gesellschaft für Mathematik und Datenverarbeitung Nr. 41, Bonn 1971.
[6] Kalmykov, S.A., Šokin, Ju. I., Juldašev.Z. H., Solving Ordinaiy Differential Equations by Interval Methods [in Russian], Doklady AN SSSR, Vol. 230, No. 6 (1976).
[7] Kalmykov, S. A., Šokin, Ju. I., Juldašev, Z. H., Methods of Interval Analysis [in Russian], Nauka, Novosibirsk 1986.
[8] Marciniak, A., Finding the Integration Intervalfor Interval Methods ofRunge-Kutta Type in Floating-Point Interval Arithmetic, Pro Dialog 10 (2000), 35^15.
[9] Marco witz, U., Fehlerabschätzung bei Anfangswertaufgaben für Systeme von gewöhnlichen Differentialgleichungen mit Anwendung auf das REENTRY-Problem, Numer. Math. 24 (1973).
[10] Marciniak, A.Jnteival Methods ofRunge-KuttaType inFloating-Point Interval Arif/jmerfc [in Polish], Technical Report RB-027/99, Poznan University of Technology, Institute of Computing Science, Poznan 1999.
[11] Moore, R. E., Inteival Analysis, Prentice-Hall, Englewood Cliffs 1966.
[12] Krupowicz, A., Numerical Methods of Initial Value Problems of Ordinaiy Differential Equations [in Polish], PWN, Warsaw 1986.
[13] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, J. Wiley & Sons, Chichester 1987.
[14] Dormand, J. R., Numerical Methods for Differential Equations. A Computational Approach, CRC Press, Boca Raton 1996.
[15] Hairer, E., Nørsett, S. P., Wanner, G., Solving Ordinaiy Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin, Heidelberg 1987.
[16] Matwiejew, N. M., Methods for Integrating Ordinaiy Differential Equations [in Polish], PWN, Warsaw 1982.
